Biomathematics Patterns: Spiral Slime
MATHEMATICAL RECREATIONS by Ian Stewart
Finding mathematics in creatures great and small
Scientific American, November 2000
This past summer I attended a conference in Portugal on the mathematics of
pattern formation, and one of the lectures reminded me of my second favorite
animal. My first favorite animal is the tiger, partly because I like its
dramatically striped fur. I like my second favorite because of its patterns,
too, but this creature isn’t as elegant as the tiger. It is the slime
mold, or more specifically, a species of cellular slime mold known as Dictyostelium
Biologists find the slime mold fascinating because it lies on the borderline
between single-celled protozoa and multicellular organisms. The slime mold
also illustrates a biological truth that the explorers of the human genome
should take to heart: it’s not just your genes that matter but what
you do with them. Despite its lowly position on the tree of life, Dictyostelium manages to create astonishingly beautiful spiral patterns. To what extent
are these patterns encoded in the slime mold’s
genes? Is there, in fact, a gene for spirals?
To answer that question, we need to know how the slime mold makes its spirals.
The pattern is actually the result of collective activity. The life cycle
of Dictyostelium begins with a microscopic spore wafting along on
the winds. If the spore happens to land on a nice, moist resting spot, it
germinates into a single-celled amoeba and starts hunting for food (mostly
bacteria). When the amoeba gets big enough, it reproduces by splitting in
two. Pretty soon there are lots of amoebas.
The artistry appears when the food runs low. The amoebas clump together,
and as they make their way toward the center of the clump they sometimes
form an elegant spiral. The crowd of amoebas gradually becomes more dense
and the spiral more tightly wound. At some point it breaks up into “streaming
patterns” that look like roots or branches extending from the center.
The streams thicken, and as more and more amoebas try to get to the same
place, they pile up in a heap known as a slug (not to be confused with the
mollusk of the same name).
The slug is a colony of amoebas, but it moves as if it were a single organism.
Once it finds a dry place, it attaches itself firmly to the ground and puts
up a long stalk. At the top of the stalk is a round blob called a fruiting
body. The amoebas in the fruiting body turn into spores and blow away on
the wind, thus continuing the cycle.
Thomas Höfer, a biophysicist at Humboldt University, Berlin, has discovered
a simple system of mathematical equations that reproduces both the slime
mold’s spirals and its streaming patterns. Cornelis J. Weijer of the
University of Dundee has shown that very similar equations can model the
movement of the slug. The main factors that determine the patterns are the
density of the amoeba population, the rate at which the amoebas produce a
chemical known as cyclic AMP and the sensitivity of individual amoebas to
this chemical. Roughly speaking, each amoeba “shouts” its presence
to its neighbors by sending out cyclic AMP. The amoebas then head in the
direction from which the shouts are loudest. The spiral pattern is a mathematical
consequence of this process. It forms when the amoebas at the center of the
clump are rotating as they send out waves of cyclic AMP.
It therefore seems that most of the slime mold’s genes simply tell
it how to be an amoeba. The genes tell the cells how to send out chemical
signals, how to sense them and how to respond to them—but the spiral
patterns they produce are not specified in the genes. Instead the patterns
emerge from the mathematical rules that the amoebas are obeying. Mathematics
may define the life cycle of the slime mold as much as genetics does.
The equations that lead to this far-reaching (and controversial) conclusion
are modifications of equations devised nearly 50 years ago by English mathematician
Alan Turing, who is best known as one of the founders of computer science.
Turing was also interested in morphogenesis—the formation and differentiation
of biological tissues and organs. In 1952 he postulated that ordered patterns
in living creatures don’t need an ordered precursor. He argued that
the patterns could arise from chemical substances called morphogens that
react with one another as they diffuse through tissue.
When Turing first published his ideas, they were purely theoretical, but
a striking example of “Turing patterns” soon appeared: the Belousov-Zhabotinsky
(BZ) chemical reaction. Russian scientist B. P. Belousov and later his compatriot
A. M. Zhabotinsky discovered that mixing just a few ordinary chemicals—including
sodium bromate, sulfuric acid and malonic acid—in a petri dish will
produce concentric rings and spirals very similar to those made by the slime
mold. Similar reactions can produce stripes, spots, dappling and many other
patterns that are common in the animal kingdom.
Nevertheless, Turing’s ideas were rejected by biologists. A major problem
with his thesis was that the patterns that appear spontaneously in the BZ
reaction are not fixed—they move across the petri dish. The same is
true for all the other Turing patterns observed by chemists. In contrast,
the patterns in most living creatures are fixed. We don’t see zebras
with moving stripes or leopards with moving spots. Turing had shown theoretically
that his equations can produce both stationary and moving patterns, but laboratory
experiments seemed to create only moving ones. Later on, the chemists discovered
why: if you carry out the reactions in a gel rather than a liquid, the patterns
become stationary. Living organisms resemble gels more than they do liquids.
But by the time this distinction became clear, biologists had lost interest
in the debate.
Mathematicians, though, continued to ponder Turing’s ideas. Although
his equations were far too simple to model real biological phenomena, they
did produce the same kinds of patterns typically seen in animals. If pigments
are deposited according to the peaks and troughs of parallel waves, you get
stripes. More complex waves produce spots. The challenge for mathematicians
was to flesh out Turing’s scheme using theoretical models that more
closely simulate the workings of biology.
In 1995 two Japanese scientists found the first convincing evidence of Turing
patterns in living things. Shigeru Kondo and Rihito Asai of Kyoto University
observed growing angelfish over several months and noticed a gradual rearrangement
of their stripes. In mammals, the skin patterns simply enlarge as the creature
grows, but in maturing angelfish new stripes are constantly forming as the
older stripes split in two. What is more, the changes can be predicted by
mathematical equations very similar to Turing’s. A computer simulation
of molecular interactions in a one-dimensional array of cells yielded a wave
pattern that closely matched the rearrangement of angelfish stripes.
The movement of the stripes is rather slow, which is why we don’t generally
notice it. But, as Galileo said, “It moves all the same.” Mathematics
changes the appearance of the angelfish just as it influences the life cycle
of the slime mold—and perhaps our own as well.